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(6)

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PROBLEM 2.7.

CHAPTER 2.

Problem 2.7
The minimum mean-square error is
Jmin = σd2 − pH R−1 p

(1)

Using the spectral theorem, we may express the correlation matrix R as
R = QΛQH
R=

M
X

λk q k qH
k

(2)

k=1

Substituting Equation (2) into Equation (1)
Jmin =σd2 −
=σd2

M
X
1 H
p qk pH qk
λ
k
k=1

M
X
1 H 2
|p qk |
−
λk
k=1

Problem 2.8
When the length of the Wiener filter is greater than the model order m, the tail end of the
tap-weight vector of the Wiener filter is zero; thus,
a
w0 = m
0
Therefore, the only possible solution for the case of an over-fitted model is
a
w0 = m
0

Problem 2.9
a)
The Wiener solution is defined by
RM aM = pM
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is derived in part b) of the problem 2.18; there it is shown that the optimum weight vector
wSN so defined is given by
wSN = R−1
v s

(1)

where s is the signal component and Rv is the correlation matrix of the noise v(n). On
the other hand, the optimum weight vector of the LCMV beamformer is defined by
R−1 s(φ)
w0 = g H
s (φ)R−1 s(φ)
∗

(2)

where s(φ) is the steering vector. In general, the formulas (1) and (2) yield different values
for the weight vector of the beamformer.

Problem 2.17
Let τi be the propagation delay, measured from the zero-time reference to the ith element
of a nonuniformly spaced array, for a plane wave arriving from a direction defined by
angle θ with respect to the perpendicular to the array. For a signal of angular frequency ω,
this delay amounts to a phase shift equal to −ωτi . Let the phase shifts for all elements of
the array be collected together in a column vector denoted by d(ω, θ). The response of a
beamformer with weight vector w to a signal (with angular frequency ω) originates from
angle θ = wH d(ω, θ). Hence, constraining the response of the array at ω and θ to some
value g involves the linear constraint
wH d(ω, θ) = g
Thus, the constraint vector d(ω, θ) serves the purpose of generalizing the idea of an LCMV
beamformer beyond simply the case of a uniformly spaced array. Everything else is the
same as before, except for the fact that the correlation matrix of the received signal is no
longer Toeplitz for the case of a nonuniformly spaced array

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The first term in (7) represents a constant. Hence, testing ln Λ against a threshold is equivalent to the test
H1

sT R−1
N u ≷ λ
H0

where λ is some threshold. Equivalently, we may write
wM L = R−1
N s
where wM L is the maximum likelihood weight vector.
The results of parts a), b), and c) show that the three criteria discussed here yield the
same optimum value for the weight vector, except for a scaling factor.

Problem 2.19
a)
Assuming the use of a noncausal Wiener filter, we write
∞
X
i=−∞

w0i r(i − k) = p(−k),

k = 0, ±1, ±2, . . . , ±∞

(1)

where the sum now extends from i = −∞ to i = ∞. Define the z-transforms:
∞
X

(c) A delay by 3 time units applied to the impulse response will make the system causal
c)
and therefore
A delay
of 3 time realizable.
units applied to the impulse response will make the system causal and
therefore realizable.
48

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